Bounds on bilinear inverse forms via Gaussian quadrature with applications

We address quadrature-based approximations of the bilinear inverse form $u^\top A^{-1} u$, where $A$ is a real symmetric positive definite matrix, and analyze properties of the Gauss, Gauss-Radau, and Gauss-Lobatto quadrature. In particular, we establish monotonicity of the bounds given by these quadrature rules, compare the tightness of these bounds, and derive associated convergence rates. To our knowledge, this is the first work to establish these properties of Gauss-type quadrature for computing bilinear inverse forms, thus filling a theoretical gap regarding this classical topic. We illustrate the empirical benefits of our theoretical results by applying quadrature to speed up two Markov Chain sampling procedures for (discrete) determinantal point processes.